Various Propositions

If the function $f(z)$ is regular, single valued and bounded in the strip $0 \leq \Re z \leq \pi$, and vanishes there at the points $z_{1}, z_{2}, z_{3}, \ldots , z_{n}, \ldots$ ($z_{n}=x_{n}+iy_{n}$) then either the series of positive terms

is convergent, or $f(z)$ is identically equal to $0$.

Assume that the coefficient $f_{1}(z)$ of the linear homogeneous differential equation

is an entire function. The necessary and sufficient condition that the general solution of this differential equation be an entire function is that the remaining coefficients $f_{2}(z), f_{3}(z), \ldots , f_{n}(z)$ are also entire functions.

Let the function $w=\varphi (z)$ be regular (possibly many-valued) in the annular region $0 < \left|z\right| < \rho$, $\rho > 0$, and satisfy there identically an equation of the form

where $F_{0}(z), F_{1}(z), \ldots , F_{l-1}(z), F_{l}(z)$ are regular in a neighbourhood of the point $z=0$. If there exists a power series $c_{0}+c_{1}z+c_{2}z^{2}+\cdots$ such that the function $(\varphi (z)-c_{0}-c_{1}z-c_{2}z^{2}-\cdots -c_{n-1}z^{n-1})z^{-n}$ remains bounded in the neighbourhood of $z=0$ for infinitely many values of $n$, then $\varphi (z)$ is regular in the neighbourhood of $z=0$, and we have $\varphi (z)=c_{0}+c_{1}z+c_{2}z^{2}+\cdots +c_{n}z^{n}+\cdots$.

Let the function $f(z)$ be regular and single-valued for $R \leq \left|z\right| < \infty$, $R > 0$. Then there exists an integer $p \geq 0$, an entire function $G(z)$ and a power series

convergent for $\left|z\right| \geq R$, such that for $R \leq \left|z\right| < \infty$ we have

If in the assumption as well as in the conclusion we replaced the inequality $R \leq \left|z\right| < \infty$ by the inequality $R < \left|z\right| < \infty$, the theorem obtained would be invalid.

Let $f(z)=f(x+iy)$ be a meromorphic periodic function of period $2\pi$ that has in the strip $0 \leq \Re z < 2\pi$ only a finite number of zeros and poles. Denote by $M(y)$ the maximum and by $\mu (y)$ the minimum of $\left|f(x+iy)\right|$ for $0 \leq x \leq 2\pi$. If it is the case that

then $f(z)$ must be a rational function of $e^{iz}$.

Let $f(z)$ be an analytic function, $z=x+iy$, and let the square of the absolute value of $f(z)$

be an algebraic function of the real variables $x$ and $y$. Then the function $f(z)$ is itself an algebraic function of $z$.

There does not exist a function which is regular analytic along the real axis and which assumes for real values of the variable every value in the interior of a fixed circle. In brief: There does not exist an analytic Peano-curve.

Find, if possible, $n+1$ analytic functions $f_{1}(z), f_{2}(z), \ldots , f_{n}(z), f(z)$, which differ from each other not merely by constant factors, are regular in a domain $\mathfrak {D}$ and for which the relation

holds in $\mathfrak {D}$.

Find, if possible, two non-constant analytic functions $f(z)$ and $g(z)$, which are regular in a domain $\mathfrak {D}$ and for which the relation

holds in $\mathfrak {D}$.

The modular graph (see Volume I, p. 130) of an entire function $g(z)$ is everywhere convex from below in just two cases: Either when

or when

where $a$ and $b$ are arbitrary complex numbers, $n$ a non-negative integer.

We say that two entire functions $f(z)$ and $g(z)$ have the same $a$-points if the two functions

are both entire. Find two distinct entire functions, which have the same $a$-, $b$- and $c$-points. (Naturally we require $b\neq c$, $c\neq a$, $a\neq b$.)

Does there exist an entire function $G(z)$ which satisfies the equations

if the sequence of numbers $a_{0}, a_{1}, a_{2}, \ldots$ is prescribed arbitrarily?

Given the numbers

for $n=1,2,3,\ldots$ subject to the conditions:

$m_{n}$ is a positive integer, however the $w_{n,k}$ are unrestricted.

Then there exists an entire function such that

for $n=0,1,2,\ldots$, $0\leq k<m_{n}$.

Suppose that $F(z)$ and $G(z)$ are entire functions that have no common zeros. Then there exist two entire functions $f(z)$ and $g(z)$ such that